statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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map {f} (hf : is_poly₂ p f) (g : R →+* S) (x y : 𝕎 R) :
map g (f x y) = f (map g x) (map g y) | begin
-- this could be turned into a tactic “macro” (taking `hf` as parameter)
-- so that applications do not have to worry about the universe issue
unfreezingI {obtain ⟨φ, hf⟩ := hf},
ext n,
simp only [map_coeff, hf, map_aeval, peval, uncurry],
apply eval₂_hom_congr (ring_hom.ext_int _ _) _ rfl,
try { ex... | lemma | witt_vector.is_poly₂.map | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons",
"ring_hom.ext_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_poly_prod (n : ℕ) : mv_polynomial (fin 2 × ℕ) ℤ | rename (prod.mk (0 : fin 2)) (witt_polynomial p ℤ n) *
rename (prod.mk (1 : fin 2)) (witt_polynomial p ℤ n) | def | witt_vector.witt_poly_prod | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"mv_polynomial",
"witt_polynomial"
] | ```
(∑ i in range n, (y.coeff i)^(p^(n-i)) * p^i.val) *
(∑ i in range n, (y.coeff i)^(p^(n-i)) * p^i.val)
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_poly_prod_vars (n : ℕ) : (witt_poly_prod p n).vars ⊆ univ ×ˢ range (n + 1) | begin
rw [witt_poly_prod],
apply subset.trans (vars_mul _ _),
refine union_subset _ _;
{ refine subset.trans (vars_rename _ _) _,
simp [witt_polynomial_vars,image_subset_iff] }
end | lemma | witt_vector.witt_poly_prod_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"witt_polynomial_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_poly_prod_remainder (n : ℕ) : mv_polynomial (fin 2 × ℕ) ℤ | ∑ i in range n, p^i * (witt_mul p i)^(p^(n-i)) | def | witt_vector.witt_poly_prod_remainder | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"mv_polynomial"
] | The "remainder term" of `witt_vector.witt_poly_prod`. See `mul_poly_of_interest_aux2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_poly_prod_remainder_vars (n : ℕ) :
(witt_poly_prod_remainder p n).vars ⊆ univ ×ˢ range n | begin
rw [witt_poly_prod_remainder],
refine subset.trans (vars_sum_subset _ _) _,
rw bUnion_subset,
intros x hx,
apply subset.trans (vars_mul _ _),
refine union_subset _ _,
{ apply subset.trans (vars_pow _ _),
have : (p : mv_polynomial (fin 2 × ℕ) ℤ) = (C (p : ℤ)),
{ simp only [int.cast_coe_nat, e... | lemma | witt_vector.witt_poly_prod_remainder_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"eq_int_cast",
"int.cast_coe_nat",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remainder (n : ℕ) : mv_polynomial (fin 2 × ℕ) ℤ | (∑ (x : ℕ) in
range (n + 1),
(rename (prod.mk 0)) ((monomial (finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x))) *
∑ (x : ℕ) in
range (n + 1),
(rename (prod.mk 1)) ((monomial (finsupp.single x (p ^ (n + 1 - x)))) (↑p ^ x)) | def | witt_vector.remainder | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"finsupp.single",
"mv_polynomial"
] | `remainder p n` represents the remainder term from `mul_poly_of_interest_aux3`.
`witt_poly_prod p (n+1)` will have variables up to `n+1`,
but `remainder` will only have variables up to `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
remainder_vars (n : ℕ) : (remainder p n).vars ⊆ univ ×ˢ range (n + 1) | begin
rw [remainder],
apply subset.trans (vars_mul _ _),
refine union_subset _ _;
{ refine subset.trans (vars_sum_subset _ _) _,
rw bUnion_subset,
intros x hx,
rw [rename_monomial, vars_monomial, finsupp.map_domain_single],
{ apply subset.trans (finsupp.support_single_subset),
simp [hx], }... | lemma | witt_vector.remainder_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"finsupp.map_domain_single",
"finsupp.support_single_subset",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
poly_of_interest (n : ℕ) : mv_polynomial (fin 2 × ℕ) ℤ | witt_mul p (n + 1) + p^(n+1) * X (0, n+1) * X (1, n+1) -
(X (0, n+1)) * rename (prod.mk (1 : fin 2)) (witt_polynomial p ℤ (n + 1)) -
(X (1, n+1)) * rename (prod.mk (0 : fin 2)) (witt_polynomial p ℤ (n + 1)) | def | witt_vector.poly_of_interest | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"mv_polynomial",
"witt_polynomial"
] | This is the polynomial whose degree we want to get a handle on. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_poly_of_interest_aux1 (n : ℕ) :
(∑ i in range (n+1), p^i * (witt_mul p i)^(p^(n-i)) : mv_polynomial (fin 2 × ℕ) ℤ) =
witt_poly_prod p n | begin
simp only [witt_poly_prod],
convert witt_structure_int_prop p (X (0 : fin 2) * X 1) n using 1,
{ simp only [witt_polynomial, witt_mul],
rw alg_hom.map_sum,
congr' 1 with i,
congr' 1,
have hsupp : (finsupp.single i (p ^ (n - i))).support = {i},
{ rw finsupp.support_eq_singleton,
sim... | lemma | witt_vector.mul_poly_of_interest_aux1 | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"alg_hom.map_sum",
"eq_int_cast",
"finsupp.single",
"finsupp.single_eq_same",
"finsupp.support_eq_singleton",
"int.cast_coe_nat",
"map_mul",
"mul_eq_mul_left_iff",
"mv_polynomial",
"pow_ne_zero",
"witt_polynomial",
"witt_structure_int_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_poly_of_interest_aux2 (n : ℕ) :
(p ^ n * witt_mul p n : mv_polynomial (fin 2 × ℕ) ℤ) + witt_poly_prod_remainder p n =
witt_poly_prod p n | begin
convert mul_poly_of_interest_aux1 p n,
rw [sum_range_succ, add_comm, nat.sub_self, pow_zero, pow_one],
refl
end | lemma | witt_vector.mul_poly_of_interest_aux2 | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"mv_polynomial",
"pow_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_poly_of_interest_aux3 (n : ℕ) :
witt_poly_prod p (n+1) =
- (p^(n+1) * X (0, n+1)) * (p^(n+1) * X (1, n+1)) +
(p^(n+1) * X (0, n+1)) * rename (prod.mk (1 : fin 2)) (witt_polynomial p ℤ (n + 1)) +
(p^(n+1) * X (1, n+1)) * rename (prod.mk (0 : fin 2)) (witt_polynomial p ℤ (n + 1)) +
remainder p n | begin
-- a useful auxiliary fact
have mvpz : (p ^ (n + 1) : mv_polynomial (fin 2 × ℕ) ℤ) = mv_polynomial.C (↑p ^ (n + 1)),
{ simp only [int.cast_coe_nat, eq_int_cast, C_pow, eq_self_iff_true] },
-- unfold definitions and peel off the last entries of the sums.
rw [witt_poly_prod, witt_polynomial, alg_hom.map_... | lemma | witt_vector.mul_poly_of_interest_aux3 | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"alg_hom.map_sum",
"eq_int_cast",
"int.cast_coe_nat",
"map_mul",
"map_pow",
"mv_polynomial",
"mv_polynomial.C",
"pow_one",
"pow_zero",
"tsub_self",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_poly_of_interest_aux4 (n : ℕ) :
(p ^ (n + 1) * witt_mul p (n + 1) : mv_polynomial (fin 2 × ℕ) ℤ) =
- (p^(n+1) * X (0, n+1)) * (p^(n+1) * X (1, n+1)) +
(p^(n+1) * X (0, n+1)) * rename (prod.mk (1 : fin 2)) (witt_polynomial p ℤ (n + 1)) +
(p^(n+1) * X (1, n+1)) * rename (prod.mk (0 : fin 2)) (witt_polynomial ... | begin
rw [← add_sub_assoc, eq_sub_iff_add_eq, mul_poly_of_interest_aux2],
exact mul_poly_of_interest_aux3 _ _
end | lemma | witt_vector.mul_poly_of_interest_aux4 | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"mv_polynomial",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_poly_of_interest_aux5 (n : ℕ) :
(p ^ (n + 1) : mv_polynomial (fin 2 × ℕ) ℤ) *
poly_of_interest p n =
(remainder p n - witt_poly_prod_remainder p (n + 1)) | begin
simp only [poly_of_interest, mul_sub, mul_add, sub_eq_iff_eq_add'],
rw mul_poly_of_interest_aux4 p n,
ring,
end | lemma | witt_vector.mul_poly_of_interest_aux5 | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"mv_polynomial",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_poly_of_interest_vars (n : ℕ) :
((p ^ (n + 1) : mv_polynomial (fin 2 × ℕ) ℤ) * poly_of_interest p n).vars ⊆
univ ×ˢ range (n + 1) | begin
rw mul_poly_of_interest_aux5,
apply subset.trans (vars_sub_subset _ _),
refine union_subset _ _,
{ apply remainder_vars },
{ apply witt_poly_prod_remainder_vars }
end | lemma | witt_vector.mul_poly_of_interest_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
poly_of_interest_vars_eq (n : ℕ) :
(poly_of_interest p n).vars =
((p ^ (n + 1) : mv_polynomial (fin 2 × ℕ) ℤ) * (witt_mul p (n + 1) +
p^(n+1) * X (0, n+1) * X (1, n+1) -
(X (0, n+1)) * rename (prod.mk (1 : fin 2)) (witt_polynomial p ℤ (n + 1)) -
(X (1, n+1)) * rename (prod.mk (0 : fin 2)) (witt_polyno... | begin
have : (p ^ (n + 1) : mv_polynomial (fin 2 × ℕ) ℤ) = C (p ^ (n + 1) : ℤ),
{ simp only [int.cast_coe_nat, eq_int_cast, C_pow, eq_self_iff_true] },
rw [poly_of_interest, this, vars_C_mul],
apply pow_ne_zero,
exact_mod_cast hp.out.ne_zero
end | lemma | witt_vector.poly_of_interest_vars_eq | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"eq_int_cast",
"int.cast_coe_nat",
"mv_polynomial",
"pow_ne_zero",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
poly_of_interest_vars (n : ℕ) : (poly_of_interest p n).vars ⊆ univ ×ˢ (range (n+1)) | by rw poly_of_interest_vars_eq; apply mul_poly_of_interest_vars | lemma | witt_vector.poly_of_interest_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
peval_poly_of_interest (n : ℕ) (x y : 𝕎 k) :
peval (poly_of_interest p n) ![λ i, x.coeff i, λ i, y.coeff i] =
(x * y).coeff (n + 1) + p^(n+1) * x.coeff (n+1) * y.coeff (n+1)
- y.coeff (n+1) * ∑ i in range (n+1+1), p^i * x.coeff i ^ (p^(n+1-i))
- x.coeff (n+1) * ∑ i in range (n+1+1), p^i * y.coeff i ^ (p^(n... | begin
simp only [poly_of_interest, peval, map_nat_cast, matrix.head_cons, map_pow,
function.uncurry_apply_pair, aeval_X,
matrix.cons_val_one, map_mul, matrix.cons_val_zero, map_sub],
rw [sub_sub, add_comm (_ * _), ← sub_sub],
have mvpz : (p : mv_polynomial ℕ ℤ) = mv_polynomial.C ↑p,
{ rw [eq_int_cast, int... | lemma | witt_vector.peval_poly_of_interest | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"eq_int_cast",
"int.cast_coe_nat",
"map_mul",
"map_nat_cast",
"map_pow",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons",
"mv_polynomial",
"mv_polynomial.C",
"mv_polynomial.eval₂_C",
"witt_polynomial_eq_sum_C_mul_X_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
peval_poly_of_interest' (n : ℕ) (x y : 𝕎 k) :
peval (poly_of_interest p n) ![λ i, x.coeff i, λ i, y.coeff i] =
(x * y).coeff (n + 1) - y.coeff (n+1) * x.coeff 0 ^ (p^(n+1))
- x.coeff (n+1) * y.coeff 0 ^ (p^(n+1)) | begin
rw peval_poly_of_interest,
have : (p : k) = 0 := char_p.cast_eq_zero (k) p,
simp only [this, add_zero, zero_mul, nat.succ_ne_zero, ne.def, not_false_iff, zero_pow'],
have sum_zero_pow_mul_pow_p : ∀ y : 𝕎 k,
∑ (x : ℕ) in range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0 ^ p ^ (n + 1),... | lemma | witt_vector.peval_poly_of_interest' | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"char_p.cast_eq_zero",
"zero_mul",
"zero_pow",
"zero_pow'"
] | The characteristic `p` version of `peval_poly_of_interest` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nth_mul_coeff' (n : ℕ) :
∃ f : truncated_witt_vector p (n+1) k → truncated_witt_vector p (n+1) k → k, ∀ (x y : 𝕎 k),
f (truncate_fun (n+1) x) (truncate_fun (n+1) y)
= (x * y).coeff (n+1) - y.coeff (n+1) * x.coeff 0 ^ (p^(n+1))
- x.coeff (n+1) * y.coeff 0 ^ (p^(n+1)) | begin
simp only [←peval_poly_of_interest'],
obtain ⟨f₀, hf₀⟩ := exists_restrict_to_vars k (poly_of_interest_vars p n),
let f : truncated_witt_vector p (n+1) k → truncated_witt_vector p (n+1) k → k,
{ intros x y,
apply f₀,
rintros ⟨a, ha⟩,
apply function.uncurry (![x, y]),
simp only [true_and, mu... | lemma | witt_vector.nth_mul_coeff' | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"multiset.mem_cons",
"multiset.mem_product",
"multiset.mem_range",
"multiset.range_succ",
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nth_mul_coeff (n : ℕ) :
∃ f : truncated_witt_vector p (n+1) k → truncated_witt_vector p (n+1) k → k, ∀ (x y : 𝕎 k),
(x * y).coeff (n+1) =
x.coeff (n+1) * y.coeff 0 ^ (p^(n+1)) + y.coeff (n+1) * x.coeff 0 ^ (p^(n+1)) +
f (truncate_fun (n+1) x) (truncate_fun (n+1) y) | begin
obtain ⟨f, hf⟩ := nth_mul_coeff' p k n,
use f,
intros x y,
rw hf x y,
ring,
end | lemma | witt_vector.nth_mul_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [
"ring",
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nth_remainder (n : ℕ) : (fin (n+1) → k) → (fin (n+1) → k) → k | classical.some (nth_mul_coeff p k n) | def | witt_vector.nth_remainder | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [] | Produces the "remainder function" of the `n+1`st coefficient, which does not depend on the `n+1`st
coefficients of the inputs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nth_remainder_spec (n : ℕ) (x y : 𝕎 k) :
(x * y).coeff (n+1) =
x.coeff (n+1) * y.coeff 0 ^ (p^(n+1)) + y.coeff (n+1) * x.coeff 0 ^ (p^(n+1)) +
nth_remainder p n (truncate_fun (n+1) x) (truncate_fun (n+1) y) | classical.some_spec (nth_mul_coeff p k n) _ _ | lemma | witt_vector.nth_remainder_spec | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_coeff.lean | [
"ring_theory.witt_vector.truncated",
"data.mv_polynomial.supported"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_mul_n : ℕ → ℕ → mv_polynomial ℕ ℤ | | 0 := 0
| (n+1) := λ k, bind₁ (function.uncurry $ ![(witt_mul_n n), X]) (witt_add p k) | def | witt_vector.witt_mul_n | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_p.lean | [
"ring_theory.witt_vector.is_poly"
] | [
"mv_polynomial"
] | `witt_mul_n p n` is the family of polynomials that computes
the coefficients of `x * n` in terms of the coefficients of the Witt vector `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_n_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) :
(x * n).coeff k = aeval x.coeff (witt_mul_n p n k) | begin
induction n with n ih generalizing k,
{ simp only [nat.nat_zero_eq_zero, nat.cast_zero, mul_zero,
zero_coeff, witt_mul_n, alg_hom.map_zero, pi.zero_apply], },
{ rw [witt_mul_n, nat.succ_eq_add_one, nat.cast_add, nat.cast_one, mul_add, mul_one,
aeval_bind₁, add_coeff],
apply eval₂_hom_congr (... | lemma | witt_vector.mul_n_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_p.lean | [
"ring_theory.witt_vector.is_poly"
] | [
"alg_hom.map_zero",
"ih",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons",
"mul_one",
"mul_zero",
"nat.cast_add",
"nat.cast_one",
"nat.cast_zero",
"ring_hom.ext_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_n_is_poly (n : ℕ) : is_poly p (λ R _Rcr x, by exactI x * n) | ⟨⟨witt_mul_n p n, λ R _Rcr x, by { funext k, exactI mul_n_coeff n x k }⟩⟩ | lemma | witt_vector.mul_n_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_p.lean | [
"ring_theory.witt_vector.is_poly"
] | [
"is_poly"
] | Multiplication by `n` is a polynomial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind₁_witt_mul_n_witt_polynomial (n k : ℕ) :
bind₁ (witt_mul_n p n) (witt_polynomial p ℤ k) = n * witt_polynomial p ℤ k | begin
induction n with n ih,
{ simp only [witt_mul_n, nat.cast_zero, zero_mul, bind₁_zero_witt_polynomial] },
{ rw [witt_mul_n, ← bind₁_bind₁, witt_add, witt_structure_int_prop],
simp only [alg_hom.map_add, nat.cast_succ, bind₁_X_right],
rw [add_mul, one_mul, bind₁_rename, bind₁_rename],
simp only [ih... | lemma | witt_vector.bind₁_witt_mul_n_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/mul_p.lean | [
"ring_theory.witt_vector.is_poly"
] | [
"alg_hom.id_apply",
"alg_hom.map_add",
"ih",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons",
"nat.cast_succ",
"nat.cast_zero",
"one_mul",
"witt_polynomial",
"witt_structure_int_prop",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_rat (Φ : mv_polynomial idx ℚ) (n : ℕ) :
mv_polynomial (idx × ℕ) ℚ | bind₁ (λ k, bind₁ (λ i, rename (prod.mk i) (W_ ℚ k)) Φ) (X_in_terms_of_W p ℚ n) | def | witt_structure_rat | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"X_in_terms_of_W",
"mv_polynomial"
] | `witt_structure_rat Φ` is a family of polynomials `ℕ → mv_polynomial (idx × ℕ) ℚ`
that are uniquely characterised by the property that
```
bind₁ (witt_structure_rat p Φ) (witt_polynomial p ℚ n) =
bind₁ (λ i, (rename (prod.mk i) (witt_polynomial p ℚ n))) Φ
```
In other words: evaluating the `n`-th Witt polynomial on the... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_structure_rat_prop (Φ : mv_polynomial idx ℚ) (n : ℕ) :
bind₁ (witt_structure_rat p Φ) (W_ ℚ n) =
bind₁ (λ i, (rename (prod.mk i) (W_ ℚ n))) Φ | calc bind₁ (witt_structure_rat p Φ) (W_ ℚ n)
= bind₁ (λ k, bind₁ (λ i, (rename (prod.mk i)) (W_ ℚ k)) Φ)
(bind₁ (X_in_terms_of_W p ℚ) (W_ ℚ n)) :
by { rw bind₁_bind₁, exact eval₂_hom_congr (ring_hom.ext_rat _ _) rfl rfl }
... = bind₁ (λ i, (rename (prod.mk i) (W_ ℚ n))) Φ :
by rw [bind₁_X_in_ter... | theorem | witt_structure_rat_prop | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"X_in_terms_of_W",
"bind₁_X_in_terms_of_W_witt_polynomial",
"mv_polynomial",
"ring_hom.ext_rat",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_rat_exists_unique (Φ : mv_polynomial idx ℚ) :
∃! (φ : ℕ → mv_polynomial (idx × ℕ) ℚ),
∀ (n : ℕ), bind₁ φ (W_ ℚ n) = bind₁ (λ i, (rename (prod.mk i) (W_ ℚ n))) Φ | begin
refine ⟨witt_structure_rat p Φ, _, _⟩,
{ intro n, apply witt_structure_rat_prop },
{ intros φ H,
funext n,
rw show φ n = bind₁ φ (bind₁ (W_ ℚ) (X_in_terms_of_W p ℚ n)),
{ rw [bind₁_witt_polynomial_X_in_terms_of_W p, bind₁_X_right] },
rw [bind₁_bind₁],
exact eval₂_hom_congr (ring_hom.... | theorem | witt_structure_rat_exists_unique | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"X_in_terms_of_W",
"bind₁_witt_polynomial_X_in_terms_of_W",
"mv_polynomial",
"ring_hom.ext_rat",
"witt_structure_rat_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_rat_rec_aux (Φ : mv_polynomial idx ℚ) (n : ℕ) :
witt_structure_rat p Φ n * C (p ^ n : ℚ) =
bind₁ (λ b, rename (λ i, (b, i)) (W_ ℚ n)) Φ -
∑ i in range n, C (p ^ i : ℚ) * (witt_structure_rat p Φ i) ^ p ^ (n - i) | begin
have := X_in_terms_of_W_aux p ℚ n,
replace := congr_arg (bind₁ (λ k : ℕ, bind₁ (λ i, rename (prod.mk i) (W_ ℚ k)) Φ)) this,
rw [alg_hom.map_mul, bind₁_C_right] at this,
rw [witt_structure_rat, this], clear this,
conv_lhs { simp only [alg_hom.map_sub, bind₁_X_right] },
rw sub_right_inj,
simp only [al... | lemma | witt_structure_rat_rec_aux | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"X_in_terms_of_W_aux",
"alg_hom.map_mul",
"alg_hom.map_pow",
"alg_hom.map_sub",
"alg_hom.map_sum",
"mv_polynomial",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_rat_rec (Φ : mv_polynomial idx ℚ) (n : ℕ) :
(witt_structure_rat p Φ n) = C (1 / p ^ n : ℚ) *
(bind₁ (λ b, (rename (λ i, (b, i)) (W_ ℚ n))) Φ -
∑ i in range n, C (p ^ i : ℚ) * (witt_structure_rat p Φ i) ^ p ^ (n - i)) | begin
calc witt_structure_rat p Φ n
= C (1 / p ^ n : ℚ) * (witt_structure_rat p Φ n * C (p ^ n : ℚ)) : _
... = _ : by rw witt_structure_rat_rec_aux,
rw [mul_left_comm, ← C_mul, div_mul_cancel, C_1, mul_one],
exact pow_ne_zero _ (nat.cast_ne_zero.2 hp.1.ne_zero),
end | lemma | witt_structure_rat_rec | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"div_mul_cancel",
"mul_left_comm",
"mul_one",
"mv_polynomial",
"pow_ne_zero",
"witt_structure_rat",
"witt_structure_rat_rec_aux"
] | Write `witt_structure_rat p φ n` in terms of `witt_structure_rat p φ i` for `i < n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_structure_int (Φ : mv_polynomial idx ℤ) (n : ℕ) :
mv_polynomial (idx × ℕ) ℤ | finsupp.map_range rat.num (rat.coe_int_num 0)
(witt_structure_rat p (map (int.cast_ring_hom ℚ) Φ) n) | def | witt_structure_int | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"finsupp.map_range",
"int.cast_ring_hom",
"mv_polynomial",
"rat.coe_int_num",
"witt_structure_rat"
] | `witt_structure_int Φ` is a family of polynomials `ℕ → mv_polynomial (idx × ℕ) ℤ`
that are uniquely characterised by the property that
```
bind₁ (witt_structure_int p Φ) (witt_polynomial p ℤ n) =
bind₁ (λ i, (rename (prod.mk i) (witt_polynomial p ℤ n))) Φ
```
In other words: evaluating the `n`-th Witt polynomial on the... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind₁_rename_expand_witt_polynomial (Φ : mv_polynomial idx ℤ) (n : ℕ)
(IH : ∀ m : ℕ, m < (n + 1) →
map (int.cast_ring_hom ℚ) (witt_structure_int p Φ m) =
witt_structure_rat p (map (int.cast_ring_hom ℚ) Φ) m) :
bind₁ (λ b, rename (λ i, (b, i)) (expand p (W_ ℤ n))) Φ =
bind₁ (λ i, expand p (witt_structu... | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [map_bind₁, map_rename, map_expand, rename_expand, map_witt_polynomial],
have key := (witt_structure_rat_prop p (map (int.cast_ring_hom ℚ) Φ) n).symm,
apply_fun expand p at key,
simp only [expand_bind₁] at key,
rw ke... | lemma | bind₁_rename_expand_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"finset.mem_range",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_polynomial",
"mv_polynomial",
"mv_polynomial.map_injective",
"witt_polynomial_vars",
"witt_structure_int",
"witt_structure_rat",
"witt_structure_rat_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_p_pow_dvd_bind₁_rename_witt_polynomial_sub_sum (Φ : mv_polynomial idx ℤ) (n : ℕ)
(IH : ∀ m : ℕ, m < n →
map (int.cast_ring_hom ℚ) (witt_structure_int p Φ m) =
witt_structure_rat p (map (int.cast_ring_hom ℚ) Φ) m) :
C ↑(p ^ n) ∣
(bind₁ (λ (b : idx), rename (λ i, (b, i)) (witt_polynomial p ℤ n)) Φ -
... | begin
cases n,
{ simp only [is_unit_one, int.coe_nat_zero, int.coe_nat_succ,
zero_add, pow_zero, C_1, is_unit.dvd] },
-- prepare a useful equation for rewriting
have key := bind₁_rename_expand_witt_polynomial Φ n IH,
apply_fun (map (int.cast_ring_hom (zmod (p ^ (n + 1))))) at key,
conv_lhs at key { s... | lemma | C_p_pow_dvd_bind₁_rename_witt_polynomial_sub_sum | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"aeval_witt_polynomial",
"bind₁_rename_expand_witt_polynomial",
"dvd_sub_pow_of_dvd_sub",
"finset.mem_range",
"int.cast_ring_hom",
"is_unit.dvd",
"is_unit_one",
"map_witt_polynomial",
"mul_dvd_mul_left",
"mv_polynomial",
"mv_polynomial.expand_zmod",
"nat.cast_mul",
"nat.cast_pow",
"nat.lt_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_witt_structure_int (Φ : mv_polynomial idx ℤ) (n : ℕ) :
map (int.cast_ring_hom ℚ) (witt_structure_int p Φ n) =
witt_structure_rat p (map (int.cast_ring_hom ℚ) Φ) n | begin
apply nat.strong_induction_on n, clear n,
intros n IH,
rw [witt_structure_int, map_map_range_eq_iff, int.coe_cast_ring_hom],
intro c,
rw [witt_structure_rat_rec, coeff_C_mul, mul_comm, mul_div_assoc', mul_one],
have sum_induction_steps : map (int.cast_ring_hom ℚ)
(∑ i in range n, C (p ^ i : ℤ) *
... | lemma | map_witt_structure_int | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"C_p_pow_dvd_bind₁_rename_witt_polynomial_sub_sum",
"eq_int_cast",
"finset.mem_range",
"int.cast_ring_hom",
"int.coe_cast_ring_hom",
"map_witt_polynomial",
"mul_comm",
"mul_div_assoc'",
"mul_one",
"mv_polynomial",
"pow_ne_zero",
"rat.denom_div_cast_eq_one_iff",
"rat.denom_eq_one_iff",
"rin... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_int_prop (Φ : mv_polynomial idx ℤ) (n) :
bind₁ (witt_structure_int p Φ) (witt_polynomial p ℤ n) =
bind₁ (λ i, rename (prod.mk i) (W_ ℤ n)) Φ | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
have := witt_structure_rat_prop p (map (int.cast_ring_hom ℚ) Φ) n,
simpa only [map_bind₁, ← eval₂_hom_map_hom, eval₂_hom_C_left, map_rename,
map_witt_polynomial, alg_hom.coe_to_ring_hom, map_witt_structure_int],
end | theorem | witt_structure_int_prop | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"alg_hom.coe_to_ring_hom",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_polynomial",
"map_witt_structure_int",
"mv_polynomial",
"mv_polynomial.map_injective",
"witt_polynomial",
"witt_structure_int",
"witt_structure_rat_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_witt_structure_int (Φ : mv_polynomial idx ℤ) (φ : ℕ → mv_polynomial (idx × ℕ) ℤ)
(h : ∀ n, bind₁ φ (witt_polynomial p ℤ n) = bind₁ (λ i, rename (prod.mk i) (W_ ℤ n)) Φ) :
φ = witt_structure_int p Φ | begin
funext k,
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
rw map_witt_structure_int,
refine congr_fun _ k,
apply unique_of_exists_unique (witt_structure_rat_exists_unique p (map (int.cast_ring_hom ℚ) Φ)),
{ intro n,
specialize h n,
apply_fun map (int.cast_ring_hom ℚ... | lemma | eq_witt_structure_int | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"alg_hom.coe_to_ring_hom",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_polynomial",
"map_witt_structure_int",
"mv_polynomial",
"mv_polynomial.map_injective",
"witt_polynomial",
"witt_structure_int",
"witt_structure_rat_exists_unique",
"witt_structure_rat_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_int_exists_unique (Φ : mv_polynomial idx ℤ) :
∃! (φ : ℕ → mv_polynomial (idx × ℕ) ℤ),
∀ (n : ℕ), bind₁ φ (witt_polynomial p ℤ n) = bind₁ (λ i : idx, (rename (prod.mk i) (W_ ℤ n))) Φ | ⟨witt_structure_int p Φ, witt_structure_int_prop _ _, eq_witt_structure_int _ _⟩ | theorem | witt_structure_int_exists_unique | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"eq_witt_structure_int",
"mv_polynomial",
"witt_polynomial",
"witt_structure_int_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_prop (Φ : mv_polynomial idx ℤ) (n) :
aeval (λ i, map (int.cast_ring_hom R) (witt_structure_int p Φ i)) (witt_polynomial p ℤ n) =
aeval (λ i, rename (prod.mk i) (W n)) Φ | begin
convert congr_arg (map (int.cast_ring_hom R)) (witt_structure_int_prop p Φ n) using 1;
rw hom_bind₁; apply eval₂_hom_congr (ring_hom.ext_int _ _) _ rfl,
{ refl },
{ simp only [map_rename, map_witt_polynomial] }
end | theorem | witt_structure_prop | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"int.cast_ring_hom",
"map_witt_polynomial",
"mv_polynomial",
"ring_hom.ext_int",
"witt_polynomial",
"witt_structure_int",
"witt_structure_int_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_int_rename {σ : Type*} (Φ : mv_polynomial idx ℤ) (f : idx → σ) (n : ℕ) :
witt_structure_int p (rename f Φ) n = rename (prod.map f id) (witt_structure_int p Φ n) | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [map_rename, map_witt_structure_int, witt_structure_rat, rename_bind₁, rename_rename,
bind₁_rename],
refl
end | lemma | witt_structure_int_rename | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial",
"mv_polynomial.map_injective",
"witt_structure_int",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_structure_rat_zero (Φ : mv_polynomial idx ℚ) :
constant_coeff (witt_structure_rat p Φ 0) = constant_coeff Φ | by simp only [witt_structure_rat, bind₁, map_aeval, X_in_terms_of_W_zero, constant_coeff_rename,
constant_coeff_witt_polynomial, aeval_X, constant_coeff_comp_algebra_map,
eval₂_hom_zero'_apply, ring_hom.id_apply] | lemma | constant_coeff_witt_structure_rat_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"X_in_terms_of_W_zero",
"constant_coeff_witt_polynomial",
"mv_polynomial",
"ring_hom.id_apply",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_structure_rat (Φ : mv_polynomial idx ℚ)
(h : constant_coeff Φ = 0) (n : ℕ) :
constant_coeff (witt_structure_rat p Φ n) = 0 | by simp only [witt_structure_rat, eval₂_hom_zero'_apply, h, bind₁, map_aeval, constant_coeff_rename,
constant_coeff_witt_polynomial, constant_coeff_comp_algebra_map, ring_hom.id_apply,
constant_coeff_X_in_terms_of_W] | lemma | constant_coeff_witt_structure_rat | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"constant_coeff_X_in_terms_of_W",
"constant_coeff_witt_polynomial",
"mv_polynomial",
"ring_hom.id_apply",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_structure_int_zero (Φ : mv_polynomial idx ℤ) :
constant_coeff (witt_structure_int p Φ 0) = constant_coeff Φ | begin
have inj : function.injective (int.cast_ring_hom ℚ),
{ intros m n, exact int.cast_inj.mp, },
apply inj,
rw [← constant_coeff_map, map_witt_structure_int,
constant_coeff_witt_structure_rat_zero, constant_coeff_map],
end | lemma | constant_coeff_witt_structure_int_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"constant_coeff_witt_structure_rat_zero",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial",
"witt_structure_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_structure_int (Φ : mv_polynomial idx ℤ)
(h : constant_coeff Φ = 0) (n : ℕ) :
constant_coeff (witt_structure_int p Φ n) = 0 | begin
have inj : function.injective (int.cast_ring_hom ℚ),
{ intros m n, exact int.cast_inj.mp, },
apply inj,
rw [← constant_coeff_map, map_witt_structure_int,
constant_coeff_witt_structure_rat, ring_hom.map_zero],
rw [constant_coeff_map, h, ring_hom.map_zero],
end | lemma | constant_coeff_witt_structure_int | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"constant_coeff_witt_structure_rat",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial",
"ring_hom.map_zero",
"witt_structure_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_rat_vars [fintype idx] (Φ : mv_polynomial idx ℚ) (n : ℕ) :
(witt_structure_rat p Φ n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | begin
rw witt_structure_rat,
intros x hx,
simp only [finset.mem_product, true_and, finset.mem_univ, finset.mem_range],
obtain ⟨k, hk, hx'⟩ := mem_vars_bind₁ _ _ hx,
obtain ⟨i, -, hx''⟩ := mem_vars_bind₁ _ _ hx',
obtain ⟨j, hj, rfl⟩ := mem_vars_rename _ _ hx'',
rw [witt_polynomial_vars, finset.mem_range] a... | lemma | witt_structure_rat_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"X_in_terms_of_W_vars_subset",
"finset.mem_product",
"finset.mem_range",
"finset.mem_univ",
"finset.range",
"finset.univ",
"fintype",
"mv_polynomial",
"witt_polynomial_vars",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_structure_int_vars [fintype idx] (Φ : mv_polynomial idx ℤ) (n : ℕ) :
(witt_structure_int p Φ n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | begin
have : function.injective (int.cast_ring_hom ℚ) := int.cast_injective,
rw [← vars_map_of_injective _ this, map_witt_structure_int],
apply witt_structure_rat_vars,
end | lemma | witt_structure_int_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/structure_polynomial.lean | [
"field_theory.finite.polynomial",
"number_theory.basic",
"ring_theory.witt_vector.witt_polynomial"
] | [
"finset.range",
"finset.univ",
"fintype",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial",
"witt_structure_int",
"witt_structure_rat_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
teichmuller_fun (r : R) : 𝕎 R | ⟨p, λ n, if n = 0 then r else 0⟩ | def | witt_vector.teichmuller_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [] | The underlying function of the monoid hom `witt_vector.teichmuller`.
The `0`-th coefficient of `teichmuller_fun p r` is `r`, and all others are `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_component_teichmuller_fun (r : R) (n : ℕ) :
ghost_component n (teichmuller_fun p r) = r ^ p ^ n | begin
rw [ghost_component_apply, aeval_witt_polynomial, finset.sum_eq_single 0,
pow_zero, one_mul, tsub_zero],
{ refl },
{ intros i hi h0,
convert mul_zero _, convert zero_pow _,
{ cases i, { contradiction }, { refl } },
{ exact pow_pos hp.1.pos _ } },
{ rw finset.mem_range, intro h, exact (h ... | lemma | witt_vector.ghost_component_teichmuller_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [
"aeval_witt_polynomial",
"finset.mem_range",
"mul_zero",
"one_mul",
"pow_pos",
"pow_zero",
"tsub_zero",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_teichmuller_fun (f : R →+* S) (r : R) :
map f (teichmuller_fun p r) = teichmuller_fun p (f r) | by { ext n, cases n, { refl }, { exact f.map_zero } } | lemma | witt_vector.map_teichmuller_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
teichmuller_mul_aux₁ (x y : mv_polynomial R ℚ) :
teichmuller_fun p (x * y) = teichmuller_fun p x * teichmuller_fun p y | begin
apply (ghost_map.bijective_of_invertible p (mv_polynomial R ℚ)).1,
rw ring_hom.map_mul,
ext1 n,
simp only [pi.mul_apply, ghost_map_apply, ghost_component_teichmuller_fun, mul_pow],
end | lemma | witt_vector.teichmuller_mul_aux₁ | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [
"mul_pow",
"mv_polynomial",
"pi.mul_apply",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
teichmuller_mul_aux₂ (x y : mv_polynomial R ℤ) :
teichmuller_fun p (x * y) = teichmuller_fun p x * teichmuller_fun p y | begin
refine map_injective (mv_polynomial.map (int.cast_ring_hom ℚ))
(mv_polynomial.map_injective _ int.cast_injective) _,
simp only [teichmuller_mul_aux₁, map_teichmuller_fun, ring_hom.map_mul]
end | lemma | witt_vector.teichmuller_mul_aux₂ | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [
"int.cast_injective",
"int.cast_ring_hom",
"mv_polynomial",
"mv_polynomial.map",
"mv_polynomial.map_injective",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
teichmuller : R →* 𝕎 R | { to_fun := teichmuller_fun p,
map_one' :=
begin
ext ⟨⟩,
{ rw one_coeff_zero, refl },
{ rw one_coeff_eq_of_pos _ _ _ (nat.succ_pos n), refl }
end,
map_mul' :=
begin
intros x y,
rcases counit_surjective R x with ⟨x, rfl⟩,
rcases counit_surjective R y with ⟨y, rfl⟩,
simp only [← map_... | def | witt_vector.teichmuller | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [
"ring_hom.map_mul"
] | The Teichmüller lift of an element of `R` to `𝕎 R`.
The `0`-th coefficient of `teichmuller p r` is `r`, and all others are `0`.
This is a monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
teichmuller_coeff_zero (r : R) :
(teichmuller p r).coeff 0 = r | rfl | lemma | witt_vector.teichmuller_coeff_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
teichmuller_coeff_pos (r : R) :
∀ (n : ℕ) (hn : 0 < n), (teichmuller p r).coeff n = 0 | | (n+1) _ := rfl. | lemma | witt_vector.teichmuller_coeff_pos | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
teichmuller_zero : teichmuller p (0:R) = 0 | by ext ⟨⟩; { rw zero_coeff, refl } | lemma | witt_vector.teichmuller_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_teichmuller (f : R →+* S) (r : R) :
map f (teichmuller p r) = teichmuller p (f r) | map_teichmuller_fun _ _ _ | lemma | witt_vector.map_teichmuller | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [] | `teichmuller` is a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_component_teichmuller (r : R) (n : ℕ) :
ghost_component n (teichmuller p r) = r ^ p ^ n | ghost_component_teichmuller_fun _ _ _ | lemma | witt_vector.ghost_component_teichmuller | ring_theory.witt_vector | src/ring_theory/witt_vector/teichmuller.lean | [
"ring_theory.witt_vector.basic"
] | [] | The `n`-th ghost component of `teichmuller p r` is `r ^ p ^ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncated_witt_vector (p : ℕ) (n : ℕ) (R : Type*) | fin n → R | def | truncated_witt_vector | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | A truncated Witt vector over `R` is a vector of elements of `R`,
i.e., the first `n` coefficients of a Witt vector.
We will define operations on this type that are compatible with the (untruncated) Witt
vector operations.
`truncated_witt_vector p n R` takes a parameter `p : ℕ` that is not used in the definition.
In pr... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (x : fin n → R) : truncated_witt_vector p n R | x | def | truncated_witt_vector.mk | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector"
] | Create a `truncated_witt_vector` from a vector `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff (i : fin n) (x : truncated_witt_vector p n R) : R | x i | def | truncated_witt_vector.coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector"
] | `x.coeff i` is the `i`th entry of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {x y : truncated_witt_vector p n R} (h : ∀ i, x.coeff i = y.coeff i) : x = y | funext h | lemma | truncated_witt_vector.ext | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {x y : truncated_witt_vector p n R} : x = y ↔ ∀ i, x.coeff i = y.coeff i | ⟨λ h i, by rw h, ext⟩ | lemma | truncated_witt_vector.ext_iff | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mk (x : fin n → R) (i : fin n) :
(mk p x).coeff i = x i | rfl | lemma | truncated_witt_vector.coeff_mk | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coeff (x : truncated_witt_vector p n R) :
mk p (λ i, x.coeff i) = x | by { ext i, rw [coeff_mk] } | lemma | truncated_witt_vector.mk_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out (x : truncated_witt_vector p n R) : 𝕎 R | witt_vector.mk p $ λ i, if h : i < n then x.coeff ⟨i, h⟩ else 0 | def | truncated_witt_vector.out | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector"
] | We can turn a truncated Witt vector `x` into a Witt vector
by setting all coefficients after `x` to be 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_out (x : truncated_witt_vector p n R) (i : fin n) :
x.out.coeff i = x.coeff i | by rw [out, witt_vector.coeff_mk, dif_pos i.is_lt, fin.eta] | lemma | truncated_witt_vector.coeff_out | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"fin.eta",
"truncated_witt_vector",
"witt_vector.coeff_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_injective : injective (@out p n R _) | begin
intros x y h,
ext i,
rw [witt_vector.ext_iff] at h,
simpa only [coeff_out] using h ↑i
end | lemma | truncated_witt_vector.out_injective | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"witt_vector.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun (x : 𝕎 R) : truncated_witt_vector p n R | truncated_witt_vector.mk p $ λ i, x.coeff i | def | witt_vector.truncate_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector",
"truncated_witt_vector.mk"
] | `truncate_fun n x` uses the first `n` entries of `x` to construct a `truncated_witt_vector`,
which has the same base `p` as `x`.
This function is bundled into a ring homomorphism in `witt_vector.truncate` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_truncate_fun (x : 𝕎 R) (i : fin n) :
(truncate_fun n x).coeff i = x.coeff i | by rw [truncate_fun, truncated_witt_vector.coeff_mk] | lemma | witt_vector.coeff_truncate_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector.coeff_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_truncate_fun (x : 𝕎 R) :
(truncate_fun n x).out = init n x | begin
ext i,
dsimp [truncated_witt_vector.out, init, select],
split_ifs with hi, swap, { refl },
rw [coeff_truncate_fun, fin.coe_mk],
end | lemma | witt_vector.out_truncate_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"fin.coe_mk",
"truncated_witt_vector.out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_out (x : truncated_witt_vector p n R) :
x.out.truncate_fun n = x | by simp only [witt_vector.truncate_fun, coeff_out, mk_coeff] | lemma | truncated_witt_vector.truncate_fun_out | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector",
"witt_vector.truncate_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_scalar : has_smul ℕ (truncated_witt_vector p n R) | ⟨λ m x, truncate_fun n (m • x.out)⟩ | instance | truncated_witt_vector.has_nat_scalar | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"has_smul",
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_int_scalar : has_smul ℤ (truncated_witt_vector p n R) | ⟨λ m x, truncate_fun n (m • x.out)⟩ | instance | truncated_witt_vector.has_int_scalar | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"has_smul",
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_pow : has_pow (truncated_witt_vector p n R) ℕ | ⟨λ x m, truncate_fun n (x.out ^ m)⟩ | instance | truncated_witt_vector.has_nat_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero (i : fin n) :
(0 : truncated_witt_vector p n R).coeff i = 0 | begin
show coeff i (truncate_fun _ 0 : truncated_witt_vector p n R) = 0,
rw [coeff_truncate_fun, witt_vector.zero_coeff],
end | lemma | truncated_witt_vector.coeff_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector",
"witt_vector.zero_coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tactic.interactive.witt_truncate_fun_tac : tactic unit | `[show _ = truncate_fun n _,
apply truncated_witt_vector.out_injective,
iterate { rw [out_truncate_fun] }] | def | tactic.interactive.witt_truncate_fun_tac | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector.out_injective"
] | A macro tactic used to prove that `truncate_fun` respects ring operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_fun_surjective :
surjective (@truncate_fun p n R) | function.right_inverse.surjective truncated_witt_vector.truncate_fun_out | lemma | witt_vector.truncate_fun_surjective | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector.truncate_fun_out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_zero : truncate_fun n (0 : 𝕎 R) = 0 | rfl | lemma | witt_vector.truncate_fun_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_one : truncate_fun n (1 : 𝕎 R) = 1 | rfl | lemma | witt_vector.truncate_fun_one | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_add (x y : 𝕎 R) :
truncate_fun n (x + y) = truncate_fun n x + truncate_fun n y | by { witt_truncate_fun_tac, rw init_add } | lemma | witt_vector.truncate_fun_add | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_mul (x y : 𝕎 R) :
truncate_fun n (x * y) = truncate_fun n x * truncate_fun n y | by { witt_truncate_fun_tac, rw init_mul } | lemma | witt_vector.truncate_fun_mul | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_neg (x : 𝕎 R) :
truncate_fun n (-x) = -truncate_fun n x | by { witt_truncate_fun_tac, rw init_neg } | lemma | witt_vector.truncate_fun_neg | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_sub (x y : 𝕎 R) :
truncate_fun n (x - y) = truncate_fun n x - truncate_fun n y | by { witt_truncate_fun_tac, rw init_sub } | lemma | witt_vector.truncate_fun_sub | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_nsmul (x : 𝕎 R) (m : ℕ) :
truncate_fun n (m • x) = m • truncate_fun n x | by { witt_truncate_fun_tac, rw init_nsmul } | lemma | witt_vector.truncate_fun_nsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_zsmul (x : 𝕎 R) (m : ℤ) :
truncate_fun n (m • x) = m • truncate_fun n x | by { witt_truncate_fun_tac, rw init_zsmul } | lemma | witt_vector.truncate_fun_zsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_pow (x : 𝕎 R) (m : ℕ) :
truncate_fun n (x ^ m) = truncate_fun n x ^ m | by { witt_truncate_fun_tac, rw init_pow } | lemma | witt_vector.truncate_fun_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_nat_cast (m : ℕ) : truncate_fun n (m : 𝕎 R) = m | rfl | lemma | witt_vector.truncate_fun_nat_cast | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_fun_int_cast (m : ℤ) : truncate_fun n (m : 𝕎 R) = m | rfl | lemma | witt_vector.truncate_fun_int_cast | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_surjective : surjective (truncate n : 𝕎 R → truncated_witt_vector p n R) | truncate_fun_surjective p n R | lemma | witt_vector.truncate_surjective | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_truncate (x : 𝕎 R) (i : fin n) :
(truncate n x).coeff i = x.coeff i | coeff_truncate_fun _ _ | lemma | witt_vector.coeff_truncate | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ker_truncate (x : 𝕎 R) :
x ∈ (@truncate p _ n R _).ker ↔ ∀ i < n, x.coeff i = 0 | begin
simp only [ring_hom.mem_ker, truncate, truncate_fun, ring_hom.coe_mk,
truncated_witt_vector.ext_iff, truncated_witt_vector.coeff_mk, coeff_zero],
exact fin.forall_iff
end | lemma | witt_vector.mem_ker_truncate | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"fin.forall_iff",
"ring_hom.coe_mk",
"ring_hom.mem_ker",
"truncated_witt_vector.coeff_mk",
"truncated_witt_vector.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_mk (f : ℕ → R) :
truncate n (mk p f) = truncated_witt_vector.mk _ (λ k, f k) | begin
ext i,
rw [coeff_truncate, coeff_mk, truncated_witt_vector.coeff_mk],
end | lemma | witt_vector.truncate_mk | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector.coeff_mk",
"truncated_witt_vector.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate {m : ℕ} (hm : n ≤ m) : truncated_witt_vector p m R →+* truncated_witt_vector p n R | ring_hom.lift_of_right_inverse (witt_vector.truncate m) out truncate_fun_out
⟨witt_vector.truncate n,
begin
intro x,
simp only [witt_vector.mem_ker_truncate],
intros h i hi,
exact h i (lt_of_lt_of_le hi hm)
end⟩ | def | truncated_witt_vector.truncate | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"ring_hom.lift_of_right_inverse",
"truncated_witt_vector",
"witt_vector.mem_ker_truncate"
] | A ring homomorphism that truncates a truncated Witt vector of length `m` to
a truncated Witt vector of length `n`, for `n ≤ m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_comp_witt_vector_truncate {m : ℕ} (hm : n ≤ m) :
(@truncate p _ n R _ m hm).comp (witt_vector.truncate m) = witt_vector.truncate n | ring_hom.lift_of_right_inverse_comp _ _ _ _ | lemma | truncated_witt_vector.truncate_comp_witt_vector_truncate | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"ring_hom.lift_of_right_inverse_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_witt_vector_truncate {m : ℕ} (hm : n ≤ m) (x : 𝕎 R) :
truncate hm (witt_vector.truncate m x) = witt_vector.truncate n x | ring_hom.lift_of_right_inverse_comp_apply _ _ _ _ _ | lemma | truncated_witt_vector.truncate_witt_vector_truncate | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"ring_hom.lift_of_right_inverse_comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_truncate {n₁ n₂ n₃ : ℕ} (h1 : n₁ ≤ n₂) (h2 : n₂ ≤ n₃)
(x : truncated_witt_vector p n₃ R) :
(truncate h1) (truncate h2 x) = truncate (h1.trans h2) x | begin
obtain ⟨x, rfl⟩ := witt_vector.truncate_surjective p n₃ R x,
simp only [truncate_witt_vector_truncate],
end | lemma | truncated_witt_vector.truncate_truncate | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"truncated_witt_vector",
"witt_vector.truncate_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_comp {n₁ n₂ n₃ : ℕ} (h1 : n₁ ≤ n₂) (h2 : n₂ ≤ n₃) :
(@truncate p _ _ R _ _ h1).comp (truncate h2) = truncate (h1.trans h2) | begin
ext1 x, simp only [truncate_truncate, function.comp_app, ring_hom.coe_comp]
end | lemma | truncated_witt_vector.truncate_comp | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"ring_hom.coe_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_surjective {m : ℕ} (hm : n ≤ m) : surjective (@truncate p _ _ R _ _ hm) | begin
intro x,
obtain ⟨x, rfl⟩ := witt_vector.truncate_surjective p _ R x,
exact ⟨witt_vector.truncate _ x, truncate_witt_vector_truncate _ _⟩
end | lemma | truncated_witt_vector.truncate_surjective | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"witt_vector.truncate_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_truncate {m : ℕ} (hm : n ≤ m) (i : fin n) (x : truncated_witt_vector p m R) :
(truncate hm x).coeff i = x.coeff (fin.cast_le hm i) | begin
obtain ⟨y, rfl⟩ := witt_vector.truncate_surjective p _ _ x,
simp only [truncate_witt_vector_truncate, witt_vector.coeff_truncate, fin.coe_cast_le],
end | lemma | truncated_witt_vector.coeff_truncate | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"fin.cast_le",
"fin.coe_cast_le",
"truncated_witt_vector",
"witt_vector.coeff_truncate",
"witt_vector.truncate_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card {R : Type*} [fintype R] :
fintype.card (truncated_witt_vector p n R) = fintype.card R ^ n | by simp only [truncated_witt_vector, fintype.card_fin, fintype.card_fun] | lemma | truncated_witt_vector.card | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"fintype",
"fintype.card",
"fintype.card_fin",
"fintype.card_fun",
"truncated_witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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